Intersection Products

Abstract

Given a regular imbedding i: X → Y of codimension d, a k-dimensional variety V, and a morphism f: V→Y, an intersection product X·V is constructed in A _k-d (W), W=f ^-1 (X). Although the case of primary interest is when f is a closed imbedding, so W = X∩V, there is significant benefit in allowing general morphisms f. Let g: W → X be the induced morphism. The normal cone C_W Vto W in V is a closed subcone of g* N _X Y, of pure dimension k. We define X·V to be the result of intersecting the k-cycle [C _W V] by the zero-section of g^*N _XY:

$$ X \cdot V = {s^ * }\left[ {{C_W}V} \right] $$

where s: W → g ^* N _X Y is the zero-section, and s* is the Gysin map constructed in Chapter 3. Alternatively X·V is the (k -d)-dimensional component of the class

$$ c\left( {{g^ * }{N_X}Y} \right) \cap s\left( {W,V} \right) $$

where s (W, V) is the Segre class of W in V.

If the k-cycle [C _w V] is written out as a sum Σm _i [C _i ],with C _i irreducible, one has a corresponding decomposition X·V =Σm _i α _i , with α_i a well-defined cycle-class on the support of C_i.

If the imbedding of W in V is regular of codimension d’, then E = g ^* N _X Y/N _w V is the quotient bundle, there is an excess intersection formula

$$ X\cdot V = {{c}_{{d - d}}}\left( E \right) \cap \left[ W \right] $$

Given i: X → Y as above, and a morphism f: Y’→Y, form the fibre square

$$ \begin{array}{*{20}{c}} {X'\mathop{{ \to Y'}}\limits^{j} } \hfill \\ {g \downarrow {{ \downarrow }^{f}}} \hfill \\ {X\mathop{{ \to Y}}\limits_{i} } \hfill \\ \end{array} $$

There are refined Gysin homomorphisms

$${{i}^{!}}:{{A}_{k}}Y\prime \to {{A}_{{k - d}}}X\prime$$

determined by the formula i^![V] = X·V for subvarieties Vof Y’.

In this chapter the fundamental properties of these intersection operations are proved. After proving that i! is well-defined on rational equivalence classes, the most important of these properties are:

• (i) Compatibility with flat pull-back (§ 6.2)

• (ii) (i)Compatibility with proper push-forward (§ 6.2)

• (iii) Commutativity (§ 6.4)

• (iv) Functoriality (§ 6.5).

For example, to calculate X • V, by (i) it suffices to calculate X • V' for any V' mapping properly and birationally to V; one may blow up V along V ∩f W to reduce to a case where the excess intersection formula applies. A particular case of (ii) is the assertion that the intersection products restrict to open subschemes: one may often compute intersection products locally. An important case of commutativity asserts that intersections may be carried out before or after specialization in a family; this will include a strong version of the “principle of continuity” in Chapter 10.

When Y' = Y, i ^!determines the (ordinary) Gysin homomorphisms

$$ {i^*}:{A_k}Y \to {A_{k - d}}X$$

Functoriality (iv) refines the statement that (j i)* = i* j* for i: X → Y, j: Y→Z regular embeddings.

More generally, if f: X→Y is a local complete intersection morphism, there are Gysin homomorphisms f*, and refined homomorphisms f ^! .These Gysin homomorphisms are used to describe the group A _* $\widetilde Y$ ,when $\widetilde Y$ is the blow-up of a scheme Y along a regularly imbedded subscheme. A new blowup formula describes the Gysin map from A _* Y to A * $\widetilde Y$ explicitly.

The rest of this book is based on this intersection product and the fundamental properties proved in § 6.1— § 6.5. As in Chap. 2, the formal properties can be motivated from topology. As we shall see in Chap. 19, a regular imbedding X SY of codimension d determines an orientation, or generalized Thom class, in H ^2d (Y, Y—X). The Gysin maps are the algebraic geometry versions of cap product by this orientation class, or with its pull-back to Y', if Y' maps to Y.